That seems to be somewhat in debate. Here's what Wikipedia currently says about the songs origins (with references cited):

[[ According to the documentary History of Rock 'n' Roll, Ben E. King had no intention of recording the song himself when he wrote it.[2] King had written the song for The Drifters, who passed on the chance to record it. It was not until after the "Spanish Harlem" recording session that he had some studio time left over. The session's producers, Jerry Leiber and Mike Stoller, asked if he had any more songs and King played "Stand by Me" on the piano for them. They liked it and called the studio musicians back in to record it.

Mike Stoller recalls it differently: "I remember arriving at our office as Jerry and Ben were working on lyrics for a new song. King had the beginnings of a melody that he was singing a capella. I went to the piano and worked up the harmonies, developing a bass pattern that became the signature of the song. Ben and Jerry quickly finished the lyrics…"[3] The fact that arranger Stan Applebaum couldn't possibly have dashed off an entire string chart at the end of a session, much less copied and distributed all the parts, supports this less dramatic version of the events.]]

It sounds to me like Ben E. King did write the original song "Stand By Me" - probably without the distinctive "signature" bass riff of the song and (which to me is an important part of the "music" of this song).   Lieber and King may well have rewritten some of the original lyrics too.    When people collaborate on songs it's not always easy to to say exactly who "wrote" what part.